I know the basis vectors of the three-dimensional lattice $\Lambda = \{\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3} \}$. I also know the equation of the plane in this 3D lattice, suppose $Ax + By + Cz = 0$. Is there a method to obtain the basis vectors of the 2D lattice $(\Lambda_{2D})$ that exists on this plane? Currently, I am thinking of generating many of the lattice points on the plane and then picking two shortest vectors in non-orthogonal directions and passing it through a lattice reduction algorithm to obtain the basis. This seems to be a trail-and-error based method and I feel that there should be a better way to tackle this problem. Any suggestions are greatly appreciated.
Thank you!
I hope that my attempt to answer this question generates some interest in this problem.
Suppose, the 3D lattice is $\mathbb{Z}^3$. Suppose the basis vectors in the 2D lattice are $\{\mathbf{x_1}, \mathbf{y_1}\} = \Lambda_{2D}$. Therefore, $Ax_1 + Bx_2 + Cx_3 = 0$ and $Ay_1 + By_2 + Cy_3 = 0$. The condition that the 2D basis vectors have to satisfy is that the area covered by the parallelogram formed by the vectors $\{\mathbf{x_1}, \mathbf{y_1}\}$ should be a minimum and greater than zero.
Since, there are infinite combinations possible, suppose the components of $\mathbf{x_1}$ are $(0, -C/m, B/m)$ (where $m = gcd(B,C)$). The area of the parallelogram is given by the norm of the cross product $\mathbf{x_1} \times \mathbf{y_1}$.
\begin{align} \mathbf{x_1} \times \mathbf{y_1} &= -\frac{(Cy_3 + By_2 )}{m}\mathbf{e_x} + \frac{By_1}{m}\mathbf{e_x} + \frac{Cy_1}{m}\mathbf{e_z} \\ &= \frac{Ay_1}{m}\mathbf{e_x} + \frac{By_1}{m}\mathbf{e_x} + \frac{Cy_1}{m}\mathbf{e_z} \end{align}
The area is $y_1^2 \left( \frac{A^2 + B^2 + C^2}{m^2}\right)$, which is proportional to $y_1$. Hence, the area is minimized if $|y_1|$ is minimized and $y_1 \neq 0$. In the sequence $y_1 = \pm1, \pm2, \ldots$, $y_1$ is chosen such that there exist integer solutions for the Diophantine equation $By_2 + Cy_3 = -Ay_1$. Therefore, $y_1$ is the smallest possible natural number such that $gcd(B,C)|Ay_1$. Once $y_1$ is obtained, $y_2$ and $y_3$ are solved for the integer solutions of $By_2 + Cy_3 = -Ay_1$. Hence, the bases may be obtained.
I was wondering if this could be generalized to any 3D lattice with basis $\Lambda_{3D} = \{\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3}\}$. I realize that I will have to get a compact form of the norm of the cross product in this new basis. I am not yet sure how that would look like. I wonder if anyone has any suggestions!